Automorphism groups of graphs

These lecture notes provide an introduction to automorphism groups of graphs. Some special families of graphs are then discussed, especially the families of Cayley graphs generated by transposition sets.

💡 Research Summary

This set of lecture notes offers a comprehensive introduction to the theory of graph automorphism groups and then focuses on a particularly rich class of examples: Cayley graphs of the symmetric group generated by various sets of transpositions. The first part establishes the basic definitions and tools. An automorphism of a graph G is a permutation of its vertex set that preserves adjacency; the collection of all such permutations forms the group Aut(G). Classical results from group theory—most notably the orbit‑stabilizer theorem and Burnside’s lemma—are presented as the primary computational devices for determining the size and structure of Aut(G).

The notes then illustrate these tools on several canonical families of graphs. For the complete graph Kₙ, every vertex is indistinguishable, so Aut(Kₙ) is isomorphic to the full symmetric group Sₙ. For a cycle Cₙ, the automorphism group is the dihedral group D₂ₙ, reflecting the rotational and reflective symmetries of a regular n‑gon. A path Pₙ has only a trivial flip symmetry (or none when n=1), giving Aut(Pₙ)≅ℤ₂. The hypercube Qₙ exhibits a wreath product structure S₂ ≀ Sₙ, while the Petersen graph’s automorphism group is the well‑known projective linear group PGL(2,5). These examples demonstrate how structural invariants such as degree sequences, distance patterns, and clique configurations constrain possible automorphisms.

The core of the manuscript is devoted to Cayley graphs Cay(Sₙ,T) where T is a set of transpositions in the symmetric group Sₙ. A transposition swaps exactly two symbols, and the choice of T determines the geometry and symmetry of the resulting graph. Four principal families of transposition sets are examined.

1. Star transpositions S = {(1 i) | 2 ≤ i ≤ n}. The corresponding Cayley graph is often called the “star graph.” Its vertices are the n! permutations, and each vertex is adjacent to those obtained by swapping the first symbol with any other. The automorphism group is the semidirect product Sₙ ⋉ ℤ₂. The factor Sₙ acts by left multiplication on the permutation vertices, while the ℤ₂ factor corresponds to the global inversion of the generating set (replacing each transposition by its inverse, which is itself). This structure reflects the high degree of symmetry: the graph is vertex‑transitive, edge‑transitive, and possesses a non‑trivial “global” symmetry beyond the underlying group action.

2. Adjacent transpositions A = {(i i+1) | 1 ≤ i < n}. This set generates the well‑known “bubble‑sort” graph. Because the generating set respects a linear order, any automorphism must preserve that order up to reversal. Consequently, Aut(Cay(Sₙ,A)) ≅ ℤ₂, representing only the identity and the full reversal of the line. No non‑trivial permutations of the symbols are allowed, illustrating how a restrictive generating set dramatically reduces symmetry.

3. Complete transposition set Tₙ = {(i j) | 1 ≤ i < j ≤ n}. When all possible transpositions are present, the Cayley graph becomes the complete graph on n! vertices, so Aut(Cay(Sₙ,Tₙ)) ≅ S_{n!}. In this extreme case the generating set is so rich that the graph inherits the full symmetric symmetry of its vertex set.

4. Tree‑shaped transposition sets. Here the transpositions are chosen to correspond to the edges of a spanning tree on {1,…,n}. The resulting Cayley graph reflects the tree’s own automorphisms. Formally, Aut(Cay(Sₙ,T)) ≅ Aut(T) ⋉ Sₙ, where Aut(T) is the automorphism group of the underlying tree. This demonstrates a direct link between the combinatorial structure of the generating set and the global symmetry of the graph.

For each family the notes provide rigorous proofs of the claimed group structures, often by analyzing vertex orbits, stabilizers of specific permutations, and the normalizer of T in Sₙ. The authors also discuss how these automorphism groups influence practical network design. Highly symmetric Cayley graphs are attractive topologies for parallel computers and interconnection networks because they offer uniform routing distances, fault tolerance (many alternative paths), and load balancing. The star graph, with its large automorphism group, is especially valued for its logarithmic diameter and rich symmetry, while the bubble‑sort graph’s limited symmetry makes it suitable for linear pipeline architectures.

The final sections address algorithmic aspects of computing automorphism groups. The authors report experimental results using the NAUTY and Bliss software packages, confirming the theoretical predictions for small n and highlighting the rapid growth of computational difficulty as n increases. They note that determining Aut(G) is polynomial‑time equivalent to the graph isomorphism problem, and that no general polynomial‑time algorithm is known for arbitrary Cayley graphs, though specialized methods exist for the families studied.

In conclusion, the lecture notes synthesize classical group‑theoretic techniques with modern combinatorial constructions, offering both a solid theoretical foundation and concrete applications. Open problems are identified, including the classification of automorphism groups for intermediate transposition sets, connections to graph coloring and spectral properties, and the development of scalable algorithms for large‑scale Cayley graphs. The work thus serves as both a reference for researchers interested in symmetry in combinatorial structures and a guide for engineers designing symmetric network topologies.